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Top page ( correct Bohr model including the two electron atom )
If you copy and paste the program source code below into a text editor, you can easily compile and run this.
(This class file name is MathMethod, so save this text editor as "MathMethod.java", and compile it.)
In this program, we first input the initial x-coordinate r1 (in MM) of electron 1 and the absolute value of the total energy E ( in eV ) of the lithium atom.
Here we use the new units, ( 1 MM = 10-14 meter, 1 SS = 10-22 second, 1 MM/SS = 108 m/s ).
From the inputted values, this program outputs the last velocity ( VX,VY in MM/SS ) and number of the de Broglie's waves ( midWN ) contained in one quarter of the orbit.
import java.util.Scanner;
class MathMethod {
public static void main(String[] args) {
Scanner stdIn=new Scanner(System.in); // input r1 and |E|
System.out.println("r1 between nucleus and electron 1 (MM)？ ");
double r=stdIn.nextDouble();
System.out.println("total energy |E| in the Li (eV) ? ");
double E=stdIn.nextDouble();
double me=9.1093826e-31; double nucle=1.1529*1.0e-26;
//rm = reduced mass of an electron
double rm=(2*me*nucle)/(2*(2*me+nucle));
double pai=3.141592653589793; double epsi=8.85418781787346e-12;
double h=6.62606896e-34; double ele=1.60217653e-19;
double ab=(ele*ele)/(4*pai*epsi); double ac=ab/rm;
double Eb=-(ele*ele*ele*ele*me)/(32*epsi*epsi*h*h); // Eb=2S energy
double Rb=(4*epsi*h*h)/(pai*ele*ele*me); // Rb=2S radius
double Rbb=Rb*1.0e14;
double Rbc=Rbb/(Math.sqrt(2));
System.out.printf("Rbb:%.5f", Rbb);
for (int i=1;i < 100;i++) { // repeat until r1=initial r1+100
double rr=r*1.0e-14; // r(MM) > rr(meter)
// disb=distance between e1S and e2S
double disb=Math.sqrt((r*r*1.0e-28)+Rb*Rb);
double poten=-(2*3*ab)/rr+ab/(2*rr)+(2*ab)/disb-(3*ab)/(Rb);
//vya= total E - potential energy - 2S kinetic energy(-Eb)
double vya=-(E*1.60217646e-19)-poten+Eb;
if (vya > 0) {
// vyb=velocity from kinetic energy
double vyb=Math.sqrt(vya/me);
double VY=vyb*1.0e-8; // change m/sec to MM/SS
double prexx=r; double VX=0.0; double WN=0.0; double preyy=0.0;
double xx,yy,vk,preVY,preWN,midWN; xx=0.0;
do {
xx=prexx+VX; yy=preyy+VY; //electron 1 position after 1SS
preVY=VY;preWN=WN ;
vk=VX*VX+VY*VY; //calculation of WN from VX,VY
// WN=number of de Broglie's waves
WN=WN+(rm*vk*1.0e-6)/h;
//calculation of VX,VY from Coulomb force
double ra=Math.sqrt(prexx*prexx+preyy*preyy);
double rb=Math.sqrt(4.0*prexx*prexx+2.0*preyy*preyy);
// rc=distance between e1 and 2Se
double rc=Math.sqrt(prexx*prexx+(Rbb*Rbb)/2+(preyy+Rbc)*(preyy+Rbc));
// change MM to meter
ra=ra*1.0e-14; rb=rb*1.0e-14; rc=rc*1.0e-14;
prexx=prexx*1.0e-14; preyy=preyy*1.0e-14;
// change velocity (VX,VY)
VX=VX+1.0e-30*ac*prexx*(-3.0/(ra*ra*ra)+2.0/(rb*rb*rb)+1.0/(rc*rc*rc));
VY=VY+1.0e-30*ac*((-3.0*preyy)/(ra*ra*ra)+preyy/(rb*rb*rb)+(preyy+Rbc*1.0e-14)/(rc*rc*rc));
prexx=xx;preyy=yy;
} while (xx >= 0); //electron has moved a half of an orbit?
if (VY > -0.0001 && VY < 0.0001) { // last VY condition
System.out.print("r1: "+r+" ");
System.out.printf("VX:%.5f", VX);
System.out.printf("preVY:%.5f", preVY);
System.out.printf("VY:%.5f", VY);
midWN=(preWN+WN)/2; System.out.printf("midWN:%.5f\n", midWN);
}} r=r+1;
}}}